/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^3), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^3, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 286 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 269 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 42 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^2, INF) (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 26 ms] (24) BEST (25) proven lower bound (26) LowerBoundPropagationProof [FINISHED, 0 ms] (27) BOUNDS(n^3, INF) (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 669 ms] (30) typed CpxTrs (31) RewriteLemmaProof [LOWER BOUND(ID), 554 ms] (32) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0) -> s(0) exp(x, s(y)) -> times(x, exp(x, y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0)) towerIter(0, y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0) -> s(0) exp(x, s(y)) -> times(x, exp(x, y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0)) towerIter(0, y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: plus(0, x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0) -> s(0) exp(x, s(y)) -> times(x, exp(x, y)) p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0)) towerIter(0, y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF). The TRS R consists of the following rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: plus, p, times, exp, towerIter, encArg They will be analysed ascendingly in the following order: p < plus plus < times plus < encArg p < times p < towerIter p < encArg times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: p, plus, times, exp, towerIter, encArg They will be analysed ascendingly in the following order: p < plus plus < times plus < encArg p < times p < towerIter p < encArg times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) Induction Base: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, 0))) ->_R^Omega(1) 0' Induction Step: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, +(n4_4, 1)))) ->_R^Omega(1) s(p(s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4)))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(c5_4)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: p, plus, times, exp, towerIter, encArg They will be analysed ascendingly in the following order: p < plus plus < times plus < encArg p < times p < towerIter p < encArg times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: plus, times, exp, towerIter, encArg They will be analysed ascendingly in the following order: plus < times plus < encArg times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, b)), rt in Omega(1 + n373_4 + n373_4^2) Induction Base: plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) ->_R^Omega(1) gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b) Induction Step: plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, 1)), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) ->_R^Omega(1) s(plus(p(s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4))), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b))) ->_L^Omega(1 + n373_4) s(plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(b, c374_4))) We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^2 for the following obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: plus, times, exp, towerIter, encArg They will be analysed ascendingly in the following order: plus < times plus < encArg times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^2, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, b)), rt in Omega(1 + n373_4 + n373_4^2) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: times, exp, towerIter, encArg They will be analysed ascendingly in the following order: times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n1343_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(*(n1343_4, b)), rt in Omega(1 + b*n1343_4 + b^2*n1343_4 + n1343_4 + n1343_4^2) Induction Base: times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) ->_R^Omega(1) 0' Induction Step: times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n1343_4, 1)), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) ->_R^Omega(1) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b), times(p(s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n1343_4))), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b))) ->_L^Omega(1 + n1343_4) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b), times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n1343_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b))) ->_IH plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(*(c1344_4, b))) ->_L^Omega(1 + b + b^2) gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(b, *(n1343_4, b))) We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3). ---------------------------------------- (24) Complex Obligation (BEST) ---------------------------------------- (25) Obligation: Proved the lower bound n^3 for the following obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, b)), rt in Omega(1 + n373_4 + n373_4^2) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: times, exp, towerIter, encArg They will be analysed ascendingly in the following order: times < exp times < encArg exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (26) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (27) BOUNDS(n^3, INF) ---------------------------------------- (28) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, b)), rt in Omega(1 + n373_4 + n373_4^2) times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n1343_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(*(n1343_4, b)), rt in Omega(1 + b*n1343_4 + b^2*n1343_4 + n1343_4 + n1343_4^2) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: exp, towerIter, encArg They will be analysed ascendingly in the following order: exp < towerIter exp < encArg towerIter < encArg ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: exp(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n2697_4))) -> *3_4, rt in Omega(n2697_4) Induction Base: exp(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, 0))) Induction Step: exp(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, +(n2697_4, 1)))) ->_R^Omega(1) times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), exp(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n2697_4)))) ->_IH times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), *3_4) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) Obligation: Innermost TRS: Rules: plus(0', x) -> x plus(s(x), y) -> s(plus(p(s(x)), y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(p(s(x)), y)) exp(x, 0') -> s(0') exp(x, s(y)) -> times(x, exp(x, y)) p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) tower(x, y) -> towerIter(x, y, s(0')) towerIter(0', y, z) -> z towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_exp(x_1, x_2)) -> exp(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_tower(x_1, x_2)) -> tower(encArg(x_1), encArg(x_2)) encArg(cons_towerIter(x_1, x_2, x_3)) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_exp(x_1, x_2) -> exp(encArg(x_1), encArg(x_2)) encode_tower(x_1, x_2) -> tower(encArg(x_1), encArg(x_2)) encode_towerIter(x_1, x_2, x_3) -> towerIter(encArg(x_1), encArg(x_2), encArg(x_3)) Types: plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter 0' :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encArg :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter cons_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_plus :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_0 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_s :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_p :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_times :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_exp :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_tower :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter encode_towerIter :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter hole_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter1_4 :: 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter Lemmas: p(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n4_4))) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n4_4), rt in Omega(1 + n4_4) plus(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n373_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n373_4, b)), rt in Omega(1 + n373_4 + n373_4^2) times(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n1343_4), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(*(n1343_4, b)), rt in Omega(1 + b*n1343_4 + b^2*n1343_4 + n1343_4 + n1343_4^2) exp(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(a), gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(1, n2697_4))) -> *3_4, rt in Omega(n2697_4) Generator Equations: gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(x)) The following defined symbols remain to be analysed: towerIter, encArg They will be analysed ascendingly in the following order: towerIter < encArg ---------------------------------------- (31) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n15112_4)) -> gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n15112_4), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(+(n15112_4, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(n15112_4))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_exp:cons_p:cons_tower:cons_towerIter2_4(c15113_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (32) BOUNDS(1, INF)